The BC-system and L-functions

In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding σ of the multiplicative group of an algebraic closure of \mathbb F_p{\mathbb {F}_p} as complex roots of unity, we construct a p-adic indecomposable representation π_σ of the integral BC-system. This construction is done using the identification of the big Witt ring of [`(\mathbb F)]_p{\bar{\mathbb F}_p} and by implementing the Artin–Hasse exponentials. The obtained representations are the p-adic analogues of the complex, extremal KMS_∞ states of the BC-system. We use the theory of p-adic L-functions to determine the partition function. Together with the analogue of the Witt construction in characteristic one, these results provide further evidence towards the construction of an analogue, for the global field of rational numbers, of the curve which provides the geometric support for the arithmetic of function fields.     

Syntomic regulators andp-adic integration I: Rigid syntomic regulators

We construct a new version of syntomic cohomology, called rigid syntomic cohomology, for smooth schemes over the ring of integers of ap-adic field. This version is more refined than previous constructions and naturally maps to most of them. We construct regulators fromK-theory into rigid syntomic cohomology. We also define a “modified” syntomic cohomology, which is better behaved in explicit computations yet is isomorphic to rigid syntomic cohomology in most cases of interest.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

Homogeneous p-adic vector bundles on abelian varieties that are analytic tori II

We investigate the temperate p-adic Riemann–Hilbert functor defined by André on abelian varieties that are analytic tori. We show that this functor induces an equivalence between the category of discrete and integral representation of the temperate fundamental group of the torus on finite dimensional \mathbb C_p{{\mathbb C}_{p}} -vector spaces as well as the category of homogeneous p-adic vector bundles on the torus.     

Prolongations in differential algebra

We develop the theory of higher prolongations of algebraic varieties over fields in arbitrary characteristic with commuting Hasse-Schmidt derivations. Prolongations were introduced by Buium in the context of fields of characteristic 0 with a single derivation. Inspired by work of Vojta, we give a new construction of higher prolongations in a more general context. Generalizing a result of Buium in characteristic 0, we prove that these prolongations are represented by a certain functor, which shows that they can be viewed as ‘twisted jet spaces.’ We give a new proof of a theorem of Moosa, Pillay and Scanlon that the prolongation functor and jet space functor commute. We also prove that the m-th prolongation and m-th jet space of a variety are differentially isomorphic by showing that their infinite prolongations are isomorphic as schemes.     

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

A <Emphasis Type="Italic">p</Emphasis>-adic RANSAC algorithm for stereo vision using Hensel lifting

A p-adic variation of the Ran(dom) Sa(mple) C(onsensus) method for solving the relative pose problem in stereo vision is developed. From two 2-adically encoded images a random sample of five pairs of corresponding points is taken, and the equations for the essential matrix are solved by lifting solutions modulo 2 to the 2-adic integers. A recently devised p-adic hierarchical classification algorithm imitiating the known LBG quantization method classifies the solutions for all the samples after having determined the number of clusters using the known intra-inter validity of clusterings. In the successful case, a cluster ranking will determine the cluster containing a 2-adic approximation to the “true” solution of the problem.     

A construction of classifying spaces for p-adic group actions

One major open problem in geometric topology is the Hilbert-Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor's construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions.     

On products in algebraic K-theory

This paper investigates the product structure in algebraic K-theory of rings. The first objective is to understand the relationships between products and the kernel of the Hurewicz homomorphism relating the algebraic K-theory of any ring to the integral homology of its linear groups. The second part of the paper is devoted to the ring of integers . Using recent results of V. Voevodsky we completely determine the products in tensored with the ring of 2-adic integers. Received: January 3, 1999.     

<Emphasis Type="Italic">p</Emphasis>-adic renormalization group solutions and the euclidean renormalization group conjectures

We discuss algebraic similarity of the Wilson’s renormalization groups in the Euclidean and p-adic spaces. Automodel Hamiltonians have identical form in both cases in the framework of perturbation theory. Fermionic p-adic model has exact renormalization group solution which generates a list of non-trivial conjectures for the Euclidean case.     

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact p-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a “generalized Robba ring” for uniform pro-p groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a selfdual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.     

On recent results of ergodic property for p-adic dynamical systems

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.     

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.     

Factorisation Theorems for T-decomposable Measures on Groups

 For a real or p-adic unipotent algebraic group G, given a T∈ Hom(G, G) and T-decomposable measure on G which is either ‘full’ or symmetric, we get a decomposition , where μ₀ is T-invariant and , and this decomposition is unique upto a shift. We also show that ν₀ is T-decomposable under some additional sufficient condition and give a counter example to justify this. We generalise the above to power bounded operators on p-adic Banach spaces. We also prove some convergence-of-types theorems on p-adic groups as well as Banach spaces. (Received 21 October 2000; in revised form 21 February 2001)     

Tambara Functors on Profinite Groups and Generalized Burnside Functors

The Tambara functor was defined by Tambara in the name of TNR-functor, to treat certain ring-valued Mackey functors on a finite group. Recently Brun revealed the importance of Tambara functors in the Witt–Burnside construction. In this article, we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshida's generalized Burnside ring functor is the first example. Consequently, we can consider a Tambara functor on any profinite group. In relation with the Witt–Burnside construction, we can give a Tambara-functor structure on Elliott's functor V _M , which generalizes the completed Burnside ring functor of Dress and Siebeneicher.     

Duality in the cohomology of crystalline local systems

Let k be a perfect field of a positive characteristic p, K-the fraction field of the ring of Witt vectors W(k) Let X be a smooth and proper scheme over W(k). We present a candidate for a cohomology theory with coefficients in crystalline local systems: p -adic étale local systems on X_K characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber X _k. We show that this cohomology satysfies a duality theorem.